Chapter 9 - Mathematics for the Life Sciences

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Chapter 9: Leslie Matrix Models & Eigenvalues
1. (9.1) Leslie Matrix Models
2. (9.2) Long Term Growth Rate (Eigenvalues)
3. (9.3) Long Term Population Structure
(Corresponding Eigenvectors)
1. (9.1) Leslie Matrix Models
Introduction
•
In the models presented and discussed in Chapters 6, 7, and
8, nothing is created or destroyed:
–
–
•
In the landscape succession examples, we never created or
destroyed land; the land merely changed from one state to another
In the disease model, there were no births or deaths; individuals
merely moved between infected and susceptible states
In this chapter we discuss models which take into account
births and deaths
1. (9.1) Leslie Matrix Models
Motivating Example: Locusts
Suppose we are studying a population of locusts and want to
know how their population changes over time:
•
Locust have three stages in their life cycle:
–
–
–
•
Not all locust eggs will survive to become adults:
–
–
•
(1) egg
(2) nymph or hopper
(3) adult
Some will die while they are still eggs
Others will die while they are hoppers
Locusts are only able to reproduce (lay eggs) during the
adult stage of their life
1. (9.1) Leslie Matrix Models
Motivating Example: Locusts
Suppose in the particular population of locusts we are studying,
each year:
•
2% of the eggs survive to become hoppers
•
5% of the hoppers survive to become adults
•
Adults die soon after they reproduce (i.e. they do not
survive to the next year)
•
Additionally, the average female adult locust will produce
1000 eggs before she dies
1. (9.1) Leslie Matrix Models
Motivating Example: Locusts
Since it is the females that reproduce, we model only the
females of the population:
•
Let x1(t) be the number of female eggs in the locust
population at time step t
•
Let x2(t) be the number of female hoppers in the locust
population at time step t
•
Let x3(t) be the number of female adults in the locust
population at time step t
1. (9.1) Leslie Matrix Models
Motivating Example: Locusts
•
We can write a system of equations to represent how this
population changes each year:
x1t  1  0.00x1t   0.00x 2 t   1000x 3 t 
x 2 t  1  0.02x1 t   x 2 t  1  0.02x1t   0.00x 2 t   0.00x 3 t 
x 3 t  1  0.05x 2 t 
x 3 t  1  0.00x1t   0.05x 2 t   0.00x 3 t 
x1t  1  1000x 3 t 
•
x1 t 1  0
x1t 
0
1000

 


0 x 2 t  or xt 1  A  xt 
x 2 t 1 0.02 0

0.05
0 
 0

x 3 t 1
 
x 3 t 



As before, we express this as a matrix equation:
•
Now, as in the previous chapters, we are able to find how
the population changes over time
1. (9.1) Leslie Matrix Models
Example 9.1 (Six Years of Locusts)
(a) How does the population of locusts change over the course
of six years if there are only 50 adults at the initial time (no
eggs and no hoppers)?
T
x
0

0
0
50
Solution: Our initial population vector is   
 . We
want to find:
x 1  Ax 0
x 2  A 2 x 0

3
x 3  A x 0
x 4   A 4 x 0
x 5  A 5 x 0
x 6  A 6 x 0
We will use MATLAB to make these calculations:

1. (9.1) Leslie Matrix Models
Example 9.1 (Six Years of Locusts)
Here I have simply modified the file “EcoSuccessionTable” to fit
our problem:
1. (9.1) Leslie Matrix Models
Example 9.1 (Six Years of Locusts)
Running this script, we get the output in the Command Window:
Interestingly, we see
that the locust
population cycles
every 3 years.
1. (9.1) Leslie Matrix Models
Example 9.1 (Six Years of Locusts)
(b) How does the population of locusts change over the course
of six years if there are 50 eggs, 100 hoppers, and 50 adults
at the initial time?
T
x
0

50
100
50
Solution: Our initial population vector is   
.
Simply modifying and running the script, we get the output:

The population still
cycles every 3 years,
but the numbers are
different.
1. (9.1) Leslie Matrix Models
Leslie Matrices
•
This type of matrix population model is known as a Leslie matrix
model. In general, Leslie matrices have the form:
f1

s1
0

M

0
•
•
•
•
f2
f3
L
0
0
L
s2 0
M O
L
O
0
L
sn1
f n 

0 
0 

M
sn 

fi is the average number of female offspring born to a female
individual per time step in class i
si is the percent
of individuals in class i that survive to class i+1,

for i < n per time step; for i = n, sn is the proportion of individuals
in class n who survive subsequent time periods
fi is the fecundity rate, and si the survival rate
Note that the first row gives the equation corresponding to the
first age class x1 (like newborn)
1. (9.1) Leslie Matrix Models
Example 9.2 (American Bison)
•
The Leslie matrix for the American Bison is given by:
 0
0
0.42


A  0.60 0
0 

0.75 0.95
 0

•
•
•
The population is divided into calves, yearlings, and adults
(ages two or more)

Thus, females who reach the age of 2 years survive an
additional year with probability 0.95 and reproduce with the
same regularity
If we start a herd with 100 adult females, what will the herd
population structure look like for the subsequent five years?
1. (9.1) Leslie Matrix Models
Example 9.2 (American Bison)
•
•
Solution: Our initial population vector is v0  0 0 100T.
Simply modifying and running the script, we get the output:

Notice that I have
added a total column.
The female population
is growing.
2. (9.2) Long Term Growth Rate (Eigenvalues)
Introduction
•
Given a Leslie matrix, we would like to find a stable stage
distribution; that is, a distribution such that the population
remains at that distribution once there
–
–
•
In the case of the locust population, this would mean that the fractions
of the population that were eggs, hoppers and adults would stay the
same through time
In Example 9.2 we saw that a Leslie matrix could model a growing
population, thus we do not necessarily want to find vectors x(t) such
that x(t+1) = x(t) as we did in Chapter 8
What we want to do is find a distribution among the stage classes
so that the proportion in each class does not change from one
time period to the next, though the overall “number” in each
class could change
–
If the overall population increased by a factor of λ, then each class
would have increased by a factor of λ if the population were at stable
stage structure
2. (9.2) Long Term Growth Rate (Eigenvalues)
Introduction
•
A population will have this property if it satisfies the
equation
x(t+1) = λx(t),
where the number λ is called an eigenvalue
If λ = 1, then the population is remaining constant over time
(this is what occurred in the transfer matrices examples)
If λ > 1 then the population is growing over time when it is
at stable stage distribution
If λ < 1, then the population is declining over time
In many contexts, λ is referred to the long-term growth rate
•
•
•
•
–
This notion applies only when the population is at a stable stage
distribution. It may take a number of years or time steps for a
population to get close to the stable stage distribution
2. (9.2) Long Term Growth Rate (Eigenvalues)
Introduction
•
For example, suppose we have a population of children and
adults and that this population has reached a stable stage
distribution and currently the population vector is [25 10]T
Then:
Population size not
25 25
changing; proportion
(λ=1):
xt 1 1 xt  1    
10 10
children to adults: 2.5
(λ=1.5):
25 37.5
xt  1  1.5  xt   1.5     
10  15 
Population size is
growing; proportion
children to adults: 2.5
(λ=0.5):
25 12.5
xt  1  0.5  xt   0.5     
10  5 
Population size is
decreasing; proportion
children to adults: 2.5



2. (9.2) Long Term Growth Rate (Eigenvalues)
Introduction
•
Let’s see if we can find these special values mathematically:
xt 1  xt   Axt   xt 
This is the equation
we want. I is the
identity matrix. It acts
like the number
 1; that is,
it is the multiplicative
identity for matrices.
 See
in-class notes from the
document camera.

 Axt   xt   0
?
A  x t   0
 A  Ixt   0
It does not make
sense to subtract
a number from a
matrix!
 det A  I  0

a
b

0
c
d

 a  d    bc  0

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